THE SYMBOL FOR DIVISION

The Anglo-American symbol for division is of 17th century origin, and has long been used on the continent of Europe to indicate subtraction. Like most elementary combinations of lines and points, the symbol is old. It was used as early as the 10th century for the word est. When written after the letter "i", it symbolized "id est." When written after the word "it", it symbolized "interest." If written after the word "divisa", for "divisa est", this might possibly have suggested its use as a symbol of division. Towards the close of the 15th century the Lombard merchants used it to indicate a half, along with similar expressions such as this one on the right.

There is also a possibility that it was used by some Italian algebrists to indicate division. In a manuscript entitled Arithmetica and Practtica by Giacomo Filippo Biodi dal Aucisco, copied in 1684, this symbol stands for division, suggesting that various forms of this kind were probably used.

The Anglo-American symbol (above top) first appeared in print in the Teutsche Algebra by Johann Heinrich Rahn (1622-1676) which appeared in Zurich in 1659. This symbol was then made known in England by the translation of Rahn's work by Dr. John Pell in London in 1688. (Smith p406)

Around the year 1200, both the Arabic writer al-Hassar, and Fibonacci (Leonardo of Pisa), symbolised division in fraction form with the use of a horizontal bar, but it is thought likely that Fibonacci adopted al-Hassar's introduction of this symbolisation.

In his Arithmetica integra (1544) Michael Stifel employed the arrangement 24 to mean 24 divided by 8. (NCTM p139)

Michael Stifel (1486?-1567) was regarded as the greatest German algebrist of the 16th century. (Cajori p140)

SMITH, D.E. "History of Mathematics" volume II, Dover Publications 1958

THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, "Historical Topics for the Mathematics Classroom", National Council of Teachers of Mathematics (USA) 1969

CAJORI, FLORIAN "A History of Mathematics", The Macmillan Company 1926

THE SYMBOL FOR INEQUALITY

Thomas Harriot (1560-1621) was an English mathematician who lived the longer part of his life in the sixteenth century but whose outstanding publication appeared in the seventeenth century. He is of special interest to Americans, because in 1585 he was sent by Sir Walter Raleigh to the new world to survey and map what was then Virginia but is now North Carolina. As a mathematician Harriot is usually considered the founder of the English school of algebraists. His great work in this field, the Artis Analyticae Praxis was published in London posthumously in 1631, and deals largely with the theory of equations. In it he makes use of these symbols above, ">" for "is greater than" (on the left), and "<" for "is less than" (on the right.)

They were not immediately accepted, for many writers preferred these symbols, which another Englishman William Oughtred (1574-1660) had suggested in the same year in the popular Clavis Mathematicae, a work on arithmetic and algebra that did much toward spreading mathematical knowledge in that country.


Isaac Barrow (1630-1677), in a book Lectiones Opticae & Geometricae (London 1674), used these symbols as follows: 
this meant "A major est quam B"

and this meant "A minor est quam B."

These symbols to the right are modern and are not international.
The symbol on the left means "is not equal to."
The middle symbol means "is not less than."
The symbol on the right stands for "is not greater than."

In the 1647 edition of Oughtred's Clavis mathematicae these somewhat analogous symbols appear for "non majus" (on the left) and "non minus" (on the right) respectively.

On the Continent these symbols, or some of their variants, apparently invented in 1734 by the French geodesist Pierre Bouguer (1698-1758), are commonly used. Bouguer was one of the French geodesists sent to Peru to measure an arc of a meridian. (Eves p251, Smith p413)

EVES, HOWARD "An introduction to the History of Mathematics," fourth edition, Holt Rinehart Winston 1976 

SMITH, D.E. "History of Mathematics" volume II, Dover Publications 1958

THE SYMBOL FOR INFINITY

John Wallis (1616-1703) was one of the most original English mathematicians of his day. He was educated for the Church at Cambridge and entered Holy Orders, but his genius was employed chiefly in the study of mathematics. The Arithmetica infinitorum, published in 1655, is his greatest work. (Cajori p183) 

This symbol for infinity is first found in print in his 1655 publication Arithmetica Infinitorum. It may have been suggested by the fact that the Romans commonly used this symbol for a thousand, just as today the word “myriad” is used for any large number, although in the Greek it meant ten thousand. The symbol was used in expressions such as, in 1695, "jam numerus incrementorum est (infinity)." (Smith p413)

The symbol for infinity, first chosen by John Wallis in 1655, stands for a concept which has given mathematicians problems since the time of the ancient Greeks. A case in point is that of Zeno of Elea (in southern Italy) who, in the 5th century BC, proposed four paradoxes which addressed whether magnitudes (lengths or numbers) are infinitely divisible or made up of a large number of small indivisible parts. (Brinkworth and Scott p80)

Wallis thought of a triangle, base length B, as composed of an infinite number of “very thin” parallelograms whose areas (from vertex to base of the triangle) form an arithmetic progression with 0 for the first term and ( A /(infinity)). B for the last term - since the last parallelogram (along the base B of the triangle) has altitude (A/(infinity)) and base B.

The area of the triangle is the sum of the arithmetic progression
O + . . . . + (A/(infinity)).B 
= (number of terms/2). (first + last term)
=(infinity/2).(0+(A/(infinity)).B)
=(infinity/2).(A/(infinity)).B
=(A-B)/2
(NCTM p413)

CAJORI, FLORIAN "A History of Mathematics", The Macmillan Company 1926

SMITH, D.E. "History of Mathematics" volume II, Dover Publications 1958

BRINKWORTH & SCOTT "The Making of Mathematics", The Australian Association of Mathematics Inc. 1994

THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, "Historical Topics for the Mathematics Classroom", National Council of Teachers of Mathematics (USA) 1969

THE SYMBOLS FOR RATIO AND PROPORTION

The symbol : to indicate ratio seems to have originated in England early in the 17th century. It appears in a text entitled Johnson’s Arithmetick ; In two Bookes (London.1633), but to indicate a fraction, three quarters being written 3:4. To indicate a ratio it appears in an astronomical work, the Harmonicon Coeleste (London, 1651), by Vincent Wing. In this work the forms A : B :: C : D and A.B :: C.D appear frequently as being equal in meaning. (Smith p406)

William Oughtred (1547-1660) was another English mathematician who wrote as follows: 
A : B = C : D as A B :: C D.
He laid extraordinary emphasis upon the use of mathematical symbols; altogether he used over 150 of them. Only 3 have come down to modern times, and one of these is this symbol for proportion. His notation for ratio and proportion was later widely used in England and on the Continent. (Cajori p157).

In his Clavis Mathematicae (1631) Oughtred used the dot to indicate either division or ratio, but in his Canones Sinuum (1657) the colon : is used for ratio. He wrote 62496 : 34295 :: 1 : 0 / 54.9- (Smith p 407)

As this notation gained ground it freed the dot . for use as the symbol for separation in decimal fractions. It is interesting to note the attitude of Leibniz (1646-1715) toward some of these symbols. On July 29, 1698, he wrote in a letter to John Bernoulli thus ".... in designating ratio I use not one point but two points, which I use at the same time, for division; thus for your dy.x :: dt.a I write dy:x = dt:a; for dy is to x as dt is to a, is indeed the same as, dy divided by x is equal to dt divided by a. From this equation follow then all the rules of proportion.” This conception of ratio and proportion was far in advance of that in contemporary arithmetics. (Cajori p158)

It is possible that Leibniz, who used : as a general symbol for division, took it from these writers, for he wrote in 1684 “x : y quod idem est ac x divis. Per y seu x/y.” 

The hypothesis that the ratio symbol : came from the symbol for division by dropping the bar has no historical basis. Since it is more international than the division symbol, it is probable that the latter symbol will gradually disappear. Various other symbols have been used to indicate division, but they have no particular interest at the present time. (Smith p407)

Ratio - the quotient of two numbers or quantities indicating their relative sizes. The ratio of a to b is written a : b or a/b. The first term is the antecedent and the second the consequent. (Daintith and Nelson p274)

The symbol :: for the equality of ratios, now giving way to the common sign for equality, was introduced by Oughtred circa 1628, for he later wrote "proportio, sive ratio aequalis ::" and a Dr. Pell gave it still more standing when he issued Rahn's algebra in English in 1668. The symbol seems to have been arbirarily chosen.

This symbol for continued proportion was used by English writers of the 17th and 18th centuries. For example it was used by Isaac Barrow (1630-1677) in his Lectiones Mathematicae (London, 1683), where he wrote "The character is made use of to signify continued Proportionals." It is still commonly seen in French textbooks. (Smith p413)

SMITH, D.E. "History of Mathematics" volume II, Dover Publications 1958

CAJORI, FLORIAN "A History of Mathematics", The Macmillan Company 1926

DAINTITH, JOHN and NELSON,R.D. "Dictionary of Mathematics", Penguin 1989